Spotting the rule in a sequence and continuing it.
Check the first differences: if they're constant, add the difference once more for the next term.
What number continues the series? 4, 11, 18, 25, __
Answer: 32
First differences are a constant +7, so the next term is 25 + 7 = 32. (uniqueness: only the arithmetic (+7) rule fits.)
Check the ratio between consecutive terms: if it's constant, multiply the last term by it.
What number continues the series? 3, 6, 12, 24, __
Answer: 48
Each term doubles (×2), so the next is 24 × 2 = 48. The 36 foil treats the last gap (+12) as additive. (uniqueness: geometric ×2 fits; arithmetic does not.)
If first differences aren't constant, check the second differences; extend the pattern of differences to get the next term.
What number continues the series? 2, 5, 10, 17, 26, __
Answer: 37
First differences are 3, 5, 7, 9 — rising by 2 each time. The next difference is 11, so 26 + 11 = 37 (the series is n²+1). (uniqueness: constant second difference; neither arithmetic nor geometric fits.)
Test the odd- and even-position terms as two separate series; continue the one the next position belongs to.
What number continues the series? 3, 20, 6, 17, 9, 14, 12, __
Answer: 11
Two interleaved series: positions 1,3,5,7 go 3,6,9,12 (+3); positions 2,4,6 go 20,17,14 (−3). The next term is in the −3 thread: 14 − 3 = 11. (uniqueness: no single arithmetic/geometric/quadratic rule fits — the interleaved split is required.)
Look for a repeating cycle of operations (e.g. ×3 then −2) and apply the next operation in the cycle.
What number continues the series? 2, 6, 4, 12, 10, 30, 28, __
Answer: 84
The operations cycle ×3 then −2: 2→6→4→12→10→30→28. The next operation is ×3, so 28 × 3 = 84. (uniqueness: no constant difference/ratio/2nd-difference fits — the ×3,−2 cycle is required.)